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Nonparametric maximum likelihood estimation of population size based on the counting distribution
Author(s) -
Böhning Dankmar,
Schön Dieter
Publication year - 2005
Publication title -
journal of the royal statistical society: series c (applied statistics)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.205
H-Index - 72
eISSN - 1467-9876
pISSN - 0035-9254
DOI - 10.1111/j.1467-9876.2005.05324.x
Subject(s) - poisson distribution , mathematics , estimator , statistics , poisson regression , quasi likelihood , population , count data , demography , sociology
Summary. The paper discusses the estimation of an unknown population size n . Suppose that an identification mechanism can identify n obs cases. The Horvitz–Thompson estimator of n adjusts this number by the inverse of 1− p 0 , where the latter is the probability of not identifying a case. When repeated counts of identifying the same case are available, we can use the counting distribution for estimating p 0 to solve the problem. Frequently, the Poisson distribution is used and, more recently, mixtures of Poisson distributions. Maximum likelihood estimation is discussed by means of the EM algorithm. For truncated Poisson mixtures, a nested EM algorithm is suggested and illustrated for several application cases. The algorithmic principles are used to show an inequality, stating that the Horvitz–Thompson estimator of n by using the mixed Poisson model is always at least as large as the estimator by using a homogeneous Poisson model. In turn, if the homogeneous Poisson model is misspecified it will, potentially strongly, underestimate the true population size. Examples from various areas illustrate this finding.